Enumeration of plane partitions by descents

TL;DR

This paper explores bijections between plane partitions and matrices, introduces new statistics with generating functions akin to MacMahon's formulas, and connects these to dual Grothendieck polynomials and longest increasing subsequences.

Contribution

It establishes a new bijection, proves a Cauchy-type identity for generalized dual Grothendieck polynomials, and introduces novel statistics on plane partitions.

Findings

New bijection between plane partitions and matrices

Proved a Cauchy-type identity for generalized dual Grothendieck polynomials

Introduced two new statistics with generating functions similar to MacMahon's formulas

Abstract

We study certain bijection between plane partitions and $\mathbb{N}$-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating functions are similar to classical MacMahon's formulas; one of these statistics is equidistributed with the usual volume. We also show natural connections with the longest increasing subsequences of words.